Probability, Top Gear and the Lottery

Top Gear recently got themselves in trouble in Argentina, over an argument about whether one of their registration plates was a reference to the Falklands War.

According to the Mirror, and I realise this may not be the strongest start,

"But bookies William Hill said the odds of coincidentally buying a car with that reg to take to Argentina were about 13 million to one."

Unfortunately the bookies' article isn't directly referenced so far as I can see, nor is the working out shown, but I'm instantly suspicious of such statements - a lot of assumptions (and hindsight) may have gone into this. If my Google-Fu is weak and you spot the direct link, please let me know.

Note that the BBC team strongly denies that the plate was bought specially.

To me, this is a classic potential case of underestimating how numbers stack up. Coincidences happen all the time. There are many plates that people can take offence at, many people buying them, many people travelling to somewhere it may have a different meaning and so on.

But in any case, the 13 million to one odds are presumably presented as a tacit statement that it couldn't have been a coincidence.

The UK lottery uses 49 balls, of which you have to guess 6 correctly. The number of ways of choosing 6 from 49 is given by a relatively simple formula.

 nCr = \frac{49!}{6!(49-6)!} = \frac{49!}{6!43!}

Let's work that out

 = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44 \times 43 \times \cdots \times 1}{6! \times 43 \times \cdots 1} = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times \cdots 1}

That turns out to be 13,983,816.

So, from a naive point of view the odds of winning the lottery are almost 14 million to 1. Yes, that is very unlikely. Famously much more unlikely than lots of other events, but it happens all the time.

"Unlikely" events are more psychologically available, we notice and remember them more, it can mean we assign significance to them which isn't deserved.

If Jeremy Clarkson, or one of the others in the team had won the lottery that would have made the news too, but it wouldn't imply a stitch up.

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